Questions: c^-1(x)=x-4

Transcript text: $c^{-1}(x)=x-4$

Solution

Solution Steps

To find the inverse of the function $ C(x) = x + 4 $, we need to determine a function $ C^{-1}(x) $ such that when applied to the output of $ C(x) $, it returns the original input $ x $. This involves solving the equation $ y = x + 4 $ for $ x $, which gives us $ x = y - 4 $. Therefore, the inverse function is $ C^{-1}(x) = x - 4 $.

Step 1: Define the Inverse Function

To find the inverse of the function $ C(x) = x + 4 $, we start by setting $ y = C(x) $. This gives us the equation: \[ y = x + 4 \] To isolate $ x $, we rearrange the equation: \[ x = y - 4 \] Thus, the inverse function is defined as: \[ C^{-1}(x) = x - 4 \]

Step 2: Calculate the Original Number

Using the inverse function, we can find the original number corresponding to an encoded number. For example, if we take $ \text{encoded\_number} = 11 $: \[ \text{original\_number} = C^{-1}(11) = 11 - 4 = 7 \]

Step 3: Interpret the Result

The original number $ 7 $ corresponds to the letter $ G $ in the alphabet, as per the given encoding scheme.